INDEX CORRELATIONS. 



41 



practical way we have determined the correlations between (a) the index 

 formed by dividing the length of a joint of a given leg by the length of 

 the cephalothorax, and (6) the latter dimension, for each segment of the 

 three legs studied. In other words, we have put for x^ length of cepha- 

 lothorax and for x^ the length of each joint of the legs taken individually. 

 The results are shown in table 20. 



Table 20. — Index correlations in the crayfish. 



From this table we see that in every case the gross correlation between 

 the index and the absolute length of the cephalothorax is positive. The 

 values of the coefficients vary greatly for the different indices. As is to 

 be expected for arithmetical reasons, the spurious correlation is in all cases 

 negative (cf. formula for p^ above). It is to be noted that the value of 

 the spurious correlation runs much more closely the same for all the 

 indices than does the gross correlation. This is, of course, what we should 

 expect a priori, from the very fact that we are dealing with a spurious 

 correlation ; that is, one whose origin is arithmetic rather than organic. 



The fact that, as shown in this table, the observed correlations between 

 index and absolute size all have the positive sign, while the spurious cor- 

 relations all have the negative sign, demonstrates the essential point at 

 issue, namely, that there is a true organic correlation between the indices 

 (measuring the proportions of the different parts) and the absolute size of 

 the body as measured by the length of the cephalothorax. There is no 

 possibility in this case of the argument being made that the interpretation 

 of the results is doubtful, because we can not be absolutely certain as to 

 what portion of the observed correlation is organic and what portion 

 spurious. Here the spurious correlation must in the nature of the case be 

 negative. But the actually observed gross correlations are in every case 

 positive. In other words, the tendency toward a negative correlation 



